A Notable Relation between N-Qubit and 2^N-1-Qubit Pauli Groups via Binary {rm LGr}(N,2N)

Employing the fact that the geometry of the N-qubit $N geq 2$ Pauli group is embodied in the structure of the symplectic polar space mathcal{W}(2N-1,2) and using properties of the Lagrangian Grassmannian {rm LGr}(N,2N) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N-qubit Pauli group and a certain subset of elements of the 2^N-1-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases N=3 (also addressed recently by Lévay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages, arXiv:1305.5689]) and N=4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space {rm PG}\(2^N-1,2\) of the 2^N-1-qubit Pauli group in terms of G-orbits, where G equiv {rm SL}(2,2)times {rm SL}(2,2)timescdotstimes {rm SL}(2,2)rtimes S_N, to decompose underlineπ\({rm LGr}(N,2N\)) into non-equivalent orbits. This leads to a partition of {rm LGr}(N,2N) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.